| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 2·11-s + 13-s − 14-s + 16-s + 3·17-s − 2·22-s − 23-s + 26-s − 28-s + 5·29-s + 7·31-s + 32-s + 3·34-s + 2·37-s − 7·41-s + 11·43-s − 2·44-s − 46-s + 8·47-s + 49-s + 52-s − 53-s − 56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.603·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.426·22-s − 0.208·23-s + 0.196·26-s − 0.188·28-s + 0.928·29-s + 1.25·31-s + 0.176·32-s + 0.514·34-s + 0.328·37-s − 1.09·41-s + 1.67·43-s − 0.301·44-s − 0.147·46-s + 1.16·47-s + 1/7·49-s + 0.138·52-s − 0.137·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.868416307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.868416307\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.529164641594109163133395432045, −7.87684307158352638620482634323, −7.08385989826608745876101870372, −6.28226116608042795172243934117, −5.64696771527081683327099375104, −4.82521297061664361880487803844, −3.99423435966835080062884690070, −3.09739555959104785121840729688, −2.34801417757247614772000096677, −0.937153127736983851226818001742,
0.937153127736983851226818001742, 2.34801417757247614772000096677, 3.09739555959104785121840729688, 3.99423435966835080062884690070, 4.82521297061664361880487803844, 5.64696771527081683327099375104, 6.28226116608042795172243934117, 7.08385989826608745876101870372, 7.87684307158352638620482634323, 8.529164641594109163133395432045
